On Different Geometric Formulations of Lagrangian Formalism

Published in Diﬀerential Geom. Appl., 10 n. 3 (1999), 225–255 Version with some corrections/improvements

On diﬀerent geometric formulations of Lagrangian formalism

Raﬀaele Vitolo1 Dept. of Mathematics “E. De Giorgi”, Universit`adi Lecce, via per Arnesano, 73100 Italy email: [email protected]

Abstract We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka’s theory of finite order variational sequences, and Vinogradov’s infinite order variational sequence associated with the C–spectral sequence. On one hand, we show that the direct limit of Krupka’s variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov’s C–spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be the Vinogradov’s infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences up to the space of Euler–Lagrange morphisms.

Key words: Fibred manifold, jet space, inﬁnite order jet space, variational bicomplex, variational sequence, spectral sequence. 1991 MSC: 58A12, 58A20, 58E30, 58G05.

1 This paper has been partially supported by Fondazione F. Severi, INdAM ‘F. Severi’ through a senior research fellowship, GNFM of CNR, MURST, University of Florence.

1 2 On formulations of Lagrangian formalism

Introduction

The theory of variational bicomplexes can be regarded as the natural geometrical set- ting for the calculus of variations [1, 2, 10, 11, 15, 19, 20, 21, 22, 23, 24]. The geometric objects which appear in the calculus of variations find a place on the vertices of a variational bicomplex, and are related by the morphisms of the bicomplex. Such morphisms are closely related to the differential of forms. Moreover, the global inverse problem is solved in this context. The purpose of this paper is to compare Krupka’s finite order formulation [10] to the infinite order formulation by Vinogradov [23, 24]. Krupka’s finite order variational sequence is produced when one quotients the de Rham sequence on a finite order jet space by means of an intrinsically defined subsequence. So, the morphisms of this bicomplex are either the differential of forms, or inclusions, or quotient morphisms. A finite order formulation of variational bicomplexes can help in keeping trace of the order of the geometric objects involved at each vertex of the bicomplex. But this yields several technical difficulties. For an intrinsic analysis of this theory, based on the structure form on jets [13] and the first variation formula [9], see [28]. The formulation of Vinogradov is carried on by means of the C–spectral sequence. This is a very general framework, by which one can formulate the variational sequences also in the case of the spaces of infinite jets of m dimensional submanifolds of a given m + n dimensional manifold. Moreover, C–spectral sequences play an important role in the theory of ordinary and partial differential equations, and in quantum mechanics and field theory. Here, we consider the variational sequence associated with the C–spectral sequence of the infinite order de Rham exact sequence. Roughly speaking, the infinite order de Rham exact sequence is made by forms on jet spaces of any order. This is a wiewpoint that allows to skip several hard technical difficulties. In fact, one has not to worry about the order of the objects or the operators. The relationship between Tulczyjew’s and Vinogradov’s formulations has been analysed in [4]. Here, we give a new finite order formulation of variational sequences using the C–spectral sequence on finite order jet spaces. The direct limit of this finite order C–spectral sequence turns out to be Vinogradov’s infinite order C–spectral sequence. Then, we evaluate the direct limit of Krupka’s variational bicomplex, finding a new infinite order variational sequence. Finally, we do a comparison of both finite and infinite order variational sequences finding that they are isomorphic up to the space of Euler–Lagrange morphisms. So, the logical scheme of this paper is summarised by the following diagram

New inﬁnite–order ≈ Vinogradov’s inﬁnite–order formulation - formulation 6 6

lim→ lim→

Krupka’s ﬁnite–order ≈ New ﬁnite–order formulation - formulation R. Vitolo 3

A final discussion has to be devoted to the language used in the paper. While Krupka’s approach is carried on by means of the language of sheaves (see, for example, [29]), Vinogradov’s approach uses an algebraic language (see [8] and references therein). Here, in order to compare the above approaches with a unique language, we found easier to left Krupka’s approach unchanged and to use a presheaf approach for the C–spectral sequence. In fact, the C–spectral sequence is defined in the category of differential groups (Appendix B), but, in our case, it can be carried on easily to presheaves. Of course, the algebraic language is a very powerful and natural tool, and in the next future much more could be said about finite order C–spectral sequences, for example in the case of jets of submanifolds. We observe that a basic introduction to spectral sequences is provided in Appendix B, in order to make the paper self–contained. We end the introduction with some mathematical conventions. In this paper, manifolds are connected and C∞, and maps between manifolds are C∞. Morphisms of fibred manifolds (and hence bundles) are morphisms over the identity of the base manifold, unless otherwise specified. We make use of definitions and results on presheaves and sheaves from [29]. In particular, we are concerned only with (pre)sheaves of IR–vector spaces, hence ‘(pre)sheaf morphism’ stands for morphism of (pre)sheaves of IR–vector spaces. We denote by SU the set of sections of a (pre)sheaf S over a topological space X defined on the open subset U ⊂ X. We recall that a sequence of (pre)sheaves over X is said to be exact if it is locally exact (see [29] for a more precise definition). If A, B are two sub(pre)sheaves of a sheaf S, then the wedge product A∧B is defined to be the sub(pre)sheaf of sections 2 of ∧S generated by wedge products of sections of A and B. We recall that a sheaf S over X is said to be soft if each section defined on a closed subset C ⊂ X can be extended to a section defined on any open subset U such that C ⊂ U. Moreover, S is said to be fine if it admits a partition of unity. A fine sheaf is also a soft sheaf. The sheaf of sections of a vector bundle is a fine sheaf, hence a soft sheaf. m Let {Sn}n∈N be a family of (pre)sheaves and {ιn : Sn → Sm}n,m∈N,n≤m be a family of injective (pre)sheaf morphisms such that, for all n,m,p ∈ N, n ≤ m ≤ p, we have p m p n ιm ◦ ιn = ιn and ιn = idSn . We say {Sn} to be an injective system. We define the direct limit of the injective system to be the (pre)sheaf

S := Sn ∼ , nG∈N

′ where ∼ is the equivalence relation deﬁned as follows. For each s ∈ Sn and s ∈ Sn′ , if ′ ′ n′ ′ n ≤ n , then s ∼ s if and only if ιn (s)= s . Acknowledgements. I would like to thank I. Kol`aˇr, D. Krupka, M. Modugno, J. Stefanekˇ and A. Vinogradov for stimulating discussions. Diagrams have been drawn by P. Taylor’s diagrams macro package. 4 On formulations of Lagrangian formalism

1 Jet spaces

In this section we recall some facts on jet spaces. We start with the definition of jet space, then we introduce the contact maps. We study the natural sheaves of forms on jet spaces which arise from the fibring and the contact maps. Finally, we introduce the horizontal and vertical differential of forms on jet spaces.

Jet spaces Our framework is a fibred manifold π : Y → X , with dim X = n and dim Y = n + m. We deal with the tangent bundle T Y → Y , the tangent prolongation Tπ : T Y → T X and the vertical bundle V Y := ker Tπ → Y . Moreover, for 0 ≤ r, we are concerned with the r–th jet space JrY ; in particular, we set J0Y ≡ Y . We recall the natural fibrings r Y Y r Y X πs : Jr → Js , π : Jr → , and the affine bundle r Y Y πr−1 : Jr → Jr−1 associated with the vector bundle

r ∗ ⊙ T X ⊗ V Y → Jr−1Y , Jr−1Y for 0 ≤ s ≤ r. A detailed account of the theory of jets can be found in [13, 11, 17]. Charts on Y adapted to the fibring are denoted by (xλ, yi). Greek indices λ,µ,... run from 1 to n and label base coordinates, Latin indices i,j,... run from 1 to m and λ i label fibre coordinates, unles otherwise specified. We denote by (∂λ,∂i) and (d ,d ), respectively, the local bases of vector fields and 1–forms on Y induced by an adapted chart. We denote multi–indices of dimension n by underlined latin letters such as p = (p1,...,pn), with 0 ≤ p1,...,pn; by identifying the index λ with a multi–index according to

λ ≃ (p1,...,pi,...,pn) ≡ (0,..., 1,..., 0) , we can write

p + λ =(p1,...,pi + 1,...,pn) .

We also set |p| := p1 + ··· + pn and p! := p1! ...pn!. Y 0 i The charts induced on Jr are denoted by (x , yp), with 0 ≤ |p| ≤ r; in particular, i i Y if |p| = 0, then we set y0 ≡ y . The local vector ﬁelds and forms of Jr induced by the p i ﬁbre coordinates are denoted by (∂i ) and (dp), 0 ≤ |p| ≤ r, 1 ≤ i ≤ m, respectively. R. Vitolo 5

Contact maps A fundamental role is played in this paper by the “contact maps” on jet spaces (see [13]). Namely, for 1 ≤ r, we consider the natural injective ﬁbred morphism over JrY → Jr−1Y

dr : JrY × T X → TJr−1Y , X and the complementary surjective ﬁbred morphism

ϑr : JrY × TJr−1Y → VJr−1Y , Jr−1Y whose coordinate expression are

λ λ j p dr = d ⊗drλ = d ⊗(∂λ + yp+λ∂j ) , 0 ≤ |p| ≤ r − 1, j p j j λ p ϑr = ϑp ⊗∂j =(dp − yp+λd )⊗∂j , 0 ≤ |p| ≤ r − 1 . We stress that

(1) dr y ϑr = ϑr y dr = 0 2 2 (2) (ϑr) = ϑr (dr) = dr

The transpose of the map ϑr is the injective ﬁbred morphism over JrY → Jr−1Y ∗ Y ∗ Y ∗ Y ϑr : Jr × V Jr−1 → T Jr−1 . Jr−1Y We have the remarkable vector subbundle ∗ Y ∗ Y ∗ Y (3) im ϑr ⊂ Jr × T Jr−1 ⊂ T Jr , Jr−1Y and, for 0 ≤ t ≤ s ≤ r, the ﬁbred inclusions Y ∗ Y ∗ ∗ (4) Jr × im ϑt ⊂ Jr × im ϑs ⊂ im ϑr . JtY JsY ∗ The above vector subbundle im ϑr yields the splitting [13]

Y ∗ Y Y ∗X ∗ (5) Jr × T Jr−1 = Jr × T ⊕ im ϑr . Jr−1Y Jr−1Y

Distinguished sheaves of forms We are concerned with some distinguished sheaves of forms on jet spaces.

Remark 1.1. The manifold Y is a diﬀerentiable retract of JrY , hence the de Rham cohomologies of Y and JrY are isomorphic. Therefore, we reduce (pre)sheaves on JrY to sheaves on Y by considering for each (pre)sheaf S on JrY the (pre)sheaf induced by S by restricting to the tube topology on JrY , i.e. , the topology generated by open sets r −1 U U Y Y of the kind (π0) ( ), with ⊂ open in . So, from now on, the (pre)sheaves of forms on JrY and the related sub(pre)sheaves will be considered as (pre)sheaves over the topological space Y of the above kind. 6 On formulations of Lagrangian formalism

Let 0 ≤ k.

k 1. First of all, for 0 ≤ r, we consider the standard sheaf Λr of k–forms on JrY

k ∗ α : JrY → ∧T JrY .

k k 2. Then, for 0 ≤ s ≤ r, we consider the sheaves H(r,s) and Hr of horizontal forms, i.e. of local ﬁbred morphisms over JrY → JsY and JrY → X of the type

k ∗ k ∗ α : JrY → ∧T JsY and β : JrY → ∧T X ,

respectively. In coordinates, if 0 < k ≤ n, then

p ...p 1 h i1 ih λh+1 λk α = αi1...i λh+1...λk dp ∧ . . . ∧ dp ∧ d ∧ . . . ∧ d , h 1 h λ1 λk β = βλ1...λk d ∧ . . . ∧ d ;

if k > n, then

p ...p 1 k−n+l i1 ik−n+l λl+1 λn α = αi1...i − λl+1...λn dp ∧ . . . ∧ dp ∧ d ∧ . . . ∧ d , k n+l 1 k−n+l

0 Here, the coordinate functions are sections of Λr, and the indices’range is 0 ≤

|pj| ≤ s, 0 ≤ h ≤ k and 0 ≤ l ≤ n. We remark that, in the coordinate expression p j of α, the indices λj are suppressed if h = k or l = n, and the indices ij are suppressed if h = 0. k k k Clearly H(r,r) = Λr and Hr = 0 for k > n. r If 0 ≤ q ≤ r and 0 ≤ t ≤ s ≤ r, then pull–back by πq yields the sheaf inclusions

k k k k k k r∗ Hq ≃ πq Hq ⊂ Hr ⊂ H(r,t) ⊂ H(r,s) ⊂ Λr , k k k k r∗ Λq ≃ πq Λq ⊂ H(r,q) ⊂ Λr .

The above inclusions are proper inclusions if t

k k 3. For 0 ≤ s < r, we consider the subsheaf C(r,s) ⊂ H(r,s) of contact forms, i.e. of local ﬁbred morphisms over JrY → JsY of the type

Y k ∗ k ∗ Y α : Jr → ∧im ϑs+1 ⊂ ∧T Js . R. Vitolo 7

k ∗ Due to the injectivity of ϑs+1, the subsheaf C(r,s) turns out to be the sheaf of k k ∗ local ﬁbred morphisms α ∈ H(r,s) which factorise as α = ∧ϑs+1 ◦α˜, through the composition k k k α˜ - ∗ ∧ϑs+1- ∗ JrY Js+1Y × ∧V JsY ∧T JsY . JsY

k Thus, α ∈ C(r,s) if and only if its coordinate expression is of the type

p ...p 1 k i1 ik α = αi1...i ϑp ∧ . . . ∧ ϑp 0 ≤ |p |,..., |p | ≤ s , k 1 k 1 k

p ...p 0 1 k with αi1...ik ∈ Λr. If r ≤ r′, s ≤ s′ and 0 ≤ s < r, 0 ≤ s′ < r′ then we have the inclusions (see (3) and (4))

k k C(r,s) ⊂ C(r′,s′) .

k k k P 4. Furthermore, we consider the subsheaf Hr ⊂ Hr of local ﬁbred morphisms α ∈ Hr such that α is a polynomial ﬁbred morphism over Jr−1Y → X of degree k. Thus, k P Y in coordinates, α ∈ Hr if and only if αλ1,...,λk : Jr → IR is a polynomial map of i degree k with respect to the coordinates yp, with |p| = r.

k k 5. Finally, we consider the subsheaf Cr ⊂ C(r+1,r) of local ﬁbred morphisms α ∈ k k C(r+1,r) such thatα ˜ projects down on JrY . Thus, in coordinates, α ∈ Cr if and p ...p 0 1 k only if αi1...ik ∈ Λr.

Main splitting

The maps dr and ϑr induce two important derivations of degree 0 (see [17]), namely the interior products by dr and ϑr

k k k k

ih ≡ idr+1 : Λr → Λr+1 , iv ≡ iϑr+1 : Λr → Λr+1 ,

∗ ∗ which make sense by taking into account the natural inclusions JrY × T X ⊂ T JrY X and VJrY ⊂ TJrY . The ﬁbred splitting (5) yields a fundamental sheaf splitting.

Lemma 1.1. We have the splitting

1 1 1 H(r+1,r) = Hr+1 ⊕ C(r+1,r) , 8 On formulations of Lagrangian formalism where the projection on the ﬁrst factor and on the second factor are given, respectively, by 1 1 1 1 H : H(r+1,r) → Hr+1 : α 7→ ihα , V : H(r+1,r) → C(r+1,r) : α 7→ ivα .

1 λ p i If α ∈ H(r+1,r) has the coordinate expression α = αλd + αi dp (0 ≤ p ≤ r), then

i p λ p i H(α)=(αλ + ypαi ) d , V (α)= αi ϑp .

1 Proposition 1.1. The above splitting of H(r+1,r) induces the splitting

k k k−l l H(r+1,r) = C (r+1,r) ∧ Hr+1 Ml=0 (see Appendix A). We recall that, in the above splitting, direct summands with l > n vanish. We set H to be the projection of the above splitting on the summand with the highest degree of the horizontal factor. Proposition 1.2. If k ≤ n, then we have k k 1 H : H → H : α 7→ kd (α) ; (r+1,r) r+1 k! r+1 if k > n, then we have k k−n n 1 H : H → C ∧ H : α 7→ k−nϑ nd (α) . (r+1,r) (r+1,r) r+1 (k − n)! n! r+1 r+1 Proof. See Appendix A. QED We set also V := Id − H to be the projection complementary to H. Remark 1.2. If k ≤ n, then we have the coordinate expression p ...p i1 ih 1 h λ1 λk H(α)= yp +λ1 ...yp +λ αi1...i λh+1...λk d ∧ . . . ∧ d , 1 h h h with 0 ≤ h ≤ k. If k > n, then we have p ...pb q ...q j1 jl 1 k−n+l 1 l H(α)= yq +λ1 ...yq +λ αi ...b i − j ...j λl+1...λn 1 l l 1 k n+l 1 l X i1 ik−n+l λ1 λn ϑp ∧ . . . ∧ ϑp ∧ d ∧ . . . ∧ d , 1 k−n+l where 0 ≤ l ≤ n and the sum is overc the subsets

j1 jl i1 ik−n+l {q . . . q } ⊂ {p . . . p } , 1 l 1 k−n+l and . . . stands for suppressed indexes (and corresponding contact forms) belonging to one of the above subsets. c R. Vitolo 9

Now, we apply the conclusion of remark 8.1 of the Appendix A to the subsheaf k k k Λr ⊂ H(r+1,r). To this aim, we want to ﬁnd the image of Λr under the projections of the above splitting. k We denote the restrictions of H, V to Λr by h,v. Next theorem is devoted to a k characterisation of the image of Λr under h. Theorem 1.1. Let 0 < k ≤ n, and denote

k k h Hr+1 := h(Λr) .

k k h P Then, we have the inclusion Hr+1 ⊂ Hr+1. k k h P Moreover, the sheaf Hr+1 admits the following characterisation: a section α ∈ Hr+1 k k h is a section of the subsheaf Hr+1 if and only if there exists a section β ∈ Λr such that

∗ ∗ (jrs) β =(jr+1s) α for each section s : X → Y . Proof. If s : X → Y is a section, then the following identities

∗ ∗ ∗ (jrs) β =(jr+1s) h(β) , (jr+1s) v(β) = 0 , yield

∗ ∗ α = h(β) ⇔ (jrs) β =(jr+1s) α

k k P QED for all α ∈ Hr+1 and β ∈ Λr.

k P Remark 1.3. It comes from the above Theorem that not any section of Hr+1 is a k k h P section of Hr+1; indeed, a section of Hr+1 in general contains ‘too many monomials’ k h with respect to a section of Hr+1. This can be seen by means of the following example. 1 Consider a one–form β ∈ Λ0. Then we have the coordinate expressions

λ i i λ β = βλd + βid , h(β)=(βλ + yλβi)d .

1 P If α ∈ H1 , then we have the coordinate expression

j µ λ α =(αλ + yµαj λ)d .

1 It is evident that, in general, there does not exists β ∈ Λr such that h(β)= α. 10 On formulations of Lagrangian formalism

Corollary 1.1. Let dim X = 1. Then we have 1 1 h P Hr+1 = Hr+1 . Proof. From the above coordinate expressions. See also [26]. QED

k Lemma 1.2. The sheaf morphisms H, V restrict on the sheaf Λr to the surjective sheaf morphisms 1 1 1 1 h h : Λr → Hr+1 , v : Λr → Cr . Proof. The restriction of H has already been studied. As for the restriction of V , 1 it is easy to see by means of a partition of the unity that it is surjective on Cr. QED Theorem 1.2. The splitting of proposition 1.1 yields the inclusion k k k−l l h Λr ⊂ C r ∧ Hr+1 , Ml=0 and the splitting projections restrict to surjective maps. Proof. In fact, for any l ≤ k the restriction of the projection k k−l l H(r+1,r) → C (r+1,r) ∧ Hr+1 k of the splitting of proposition 1.1 to the sheaf Λr takes the form k k−l l k−l l h Λr → C r ∧ Hr+1 ⊂ C (r+1,r) ∧ Hr+1 . The above inclusion can be tested in coordinates. For the sake of simplicity, let us k−l l h consider a global section α ∈ C r ∧ Hr+1 where 0 ≤ l ≤ n. We have the coordinate expression p ...p q ...q j1 jh 1 k−l 1 h α = yq +λ1 ...yq +λ αi1 ... i − j1...j λh+1...λl 1 h h k l h i1 ik−l λ1 λl ϑp ∧ . . . ∧ ϑp ∧ d ∧ . . . ∧ d , 1 k−l 0 where 0 ≤ |pi|, |qi| ≤ r and 0 ≤ h ≤ n. If {ψi} is a partition of the unity on Λr subordinate to a coordinate atlas, let

s1...sr t1 tr λr+1 λk α˜i := ψi α˜t1...tr λr+1...λk ds ∧ . . . ∧ dp ∧ d ∧ . . . ∧ d , 1 r where the set of pairs of indices {t1 . . . tr } is a permutation of the set of pairs of indices s1 sr k k−l l − i1 ik l j1 jl h {p . . . p q . . . q }. Then α˜i is a global section of Λr whose projection on C r ∧ Hr+1 1 k−l 1 l i is α. P The proof is analogous for k > n. QED We remark that, in general, the above inclusion is a proper inclusion: in general, a k sum of elements of the direct summands is not an element of Λr. R. Vitolo 11

k Corollary 1.2. The sheaf morphism H restricts on the sheaf Λr to the surjective sheaf morphisms

k k k k−n n h h h : Λr → Hr+1 k ≤ n, h : Λr → C r ∧ Hr+1 k>n.

Horizontal and vertical diﬀerential

The derivations ih, iv, and the exterior differential d yield two derivations of degree one (see [17]). Namely, we define the horizontal and vertical differential to be the sheaf morphisms

k k k k dh := ih ◦d − d◦ih : Λr → Λr+1 , dv := iv ◦d − d◦iv : Λr → Λr+1 ,

It can be proved (see [17]) that dh and dv fulﬁll the properties

2 2 dh = dv = 0 , dh ◦dv + dv ◦dh = 0 , r+1 ∗ dh + dv =(πr ) ◦d , ∗ ∗ ∗ (jr+1s) ◦dv = 0 , d◦(jrs) =(jr+1s) ◦dh .

The action of dh and dv on functions f : JrY → IR and one–forms on JrY uniquely characterises dh and dv. We have the coordinate expressions

λ i p λ dhf =(dr+1)λ.fd =(∂λf + yp+λ∂i f)d , λ i i λ i i λ dhd = 0 , dhdp = −dp+λ ∧ d , dhϑp = −ϑp+λ ∧ d , p i dvf = ∂i fϑp , λ i i λ i dvd = 0 , dvdp = dp+λ ∧ d , dvϑp = 0 . We note that

i λ i λ i µ λ i λ −dp+λ ∧ d = −ϑp+λ ∧ d + yp+λ+µd ∧ d = −ϑp+λ ∧ d .

Finally, next Proposition analyses the relationship of dh and dv with the splitting of Proposition 1.1. Proposition 1.3. We have

k k+1 k 1 k d H ⊂ H , d H ⊂ C ∧ H , h r r+1 v r r r k h k h+1 k n d C − ∧ H ⊂ C ∧ H , d C − ∧ H = {0} , h (r,r 1) r (r+1,r) r+1 h (r,r 1) r k k+1 k k+1 d C − ⊂ C , d C ⊂ C , v (r,r 1) r v r r

Proof. From the action of dh, dv on functions and local coordinate bases of forms. QED 12 On formulations of Lagrangian formalism

Direct limit

r The sheaf injections πs (r ≥ s) provide several inclusions between the sheaves of forms previously introduced. This yields several injective systems, whose direct limit is studied here. We deﬁne the presheaves on Y

k k (k,h) (k,h) Λ := lim Λr , Λ := lim Λ (r+1,r) . → →

By simple counterexamples, it can be proved that the above presheaves are not sheaves in general, because the gluing axiom fails to be true.

k Remark 1.4. For any equivalence class [α] ∈ Λ there exists a distinguished represen- k (0,k) (k,0) tative β ∈ Λr whose order r is minimal. The same holds for Λ and Λ . Accordingly, k we shall often indicate by β ∈ Λ (without brackets) such a minimal section.

k k k Lemma 1.3. We have lim Λ(r+1,r) = lim Λr ≡ Λ. → →

k k k Proof. In fact, we have the inclusions Λr ⊂ Λ(r+1,r) ⊂ Λr+1 QED

Theorem 1.3. We have the natural splitting

k k (k−l,l) Λ= Λ . Ml=0

Proof. It comes from the above lemma and the splitting of proposition 1.1. QED

Remark 1.5. The above splitting represents one of the major differencies between the finite order and the infinite order case. As we shall see, in the infinite order formulations k one has to deal with quotients of Λ by sheaves of contact forms. The above splitting allows us to identify such quotients with ‘more concrete’ spaces (see proposition 4.2). The situation is much more complicated in the finite order case for the lack of such a k k splitting. In fact, the inclusion Λr ⊂ Λ(r+1,r) is a proper inclusion, and we are in the bad situation described in remark 8.1. Nevertheless, by means of the splitting of proposition 1.1, we are able to recover in the finite order case almost all features of infinite order formulations, but in a much more difficult way (see [28]).

k Proposition 1.4. The sheaf morphisms d, dh, dv, h, admit direct limits. Namely, such R. Vitolo 13 direct limits turn out to be the presheaf morphisms

k k+1 d : Λ → Λ : [α] 7→ [dα] , k k+1 k k+1 dh : Λ → Λ : [α] 7→ [dhα] , dv : Λ → Λ : [α] 7→ [dvα] , (0,n) k k k Λ r+1 : [α] 7→ [h] if k ≤ n h : Λ(r+1,r) → (k−n,n)  k Λ r+1 : [α] 7→ [h] if k > n ;  k Note that the map h of the above proposition turns out to be the projection of the splitting of theorem 1.3 on the factor with the highest horizontal degree; in other words, the direct limit of the projection is the projection of the splitting of the direct limit. We observe that we did not indicate the degree of d, dh and dv. This is both for a matter of ‘tradition’ and not to make too heavy the notation. Finally, next proposition analyses the relationship of dh and dv with the splitting of the above theorem.

Proposition 1.5. We have

(0,k) (0,k+1) (0,k) (1,k) dh( Λ ) ⊂ Λ , dv( Λ ) ⊂ Λ , (k,0) (k,1) (k,0) (k+1,0) dh( Λ ) ⊂ Λ , dv( Λ ) ⊂ Λ .

Proof. From the action of dh, dv on functions and local coordinate bases of forms. QED

2 Finite order variational sequence

In this section, we recall the theory of variational sequences on ﬁnite order jet bundles [10]. We give a concise summary of the theory using our notation. We consider the de Rham exact sheaf sequence on JrY

0 1 J - - d - d - d - d - 0 IR Λr Λr . . . Λr 0, where J := dim JrY . We are able to provide several natural subsequences of the de Rham sequence. For example, natural subsequences of the de Rham sequence arise k 1 1 by considering the ideals generated in Λr by its natural subsheaves H(r,s), C(r,s), . . . Not all natural subsequences of the de Rham sequence turn out to be exact. Here, we introduce an exact natural subsequence of the de Rham sequence, which is of particular importance in the variational calculus, although being deﬁned independently (see [10, 26]). 14 On formulations of Lagrangian formalism

k We introduce a new subsheaf of Λr. Namely, we set

k k ∗ CΛr = {α ∈ Λr | (jrs) α = 0 for every section s : X → Y } .

n The above subsheaf CΛr is made by forms which does not give contribution to action–like functionals. [10, 17, 27]. Lemma 2.1. We have

k k k CΛr = ker h if 0 ≤ k ≤ n , CΛr = Λr if k>n. k Proof. Let α ∈ Λr. Then, for any section s : X → Y we have

∗ ∗ (jrs) α =(jr+1s) h(α) ,

k k and α ∈ ker h implies α ∈ CΛr. Conversely, suppose α ∈ CΛr. Then we have

∗ λ λ (jr+1s) h(α)= h(α)λ1...λk ◦jr+1s d1 ∧ . . . ∧ dk , hence h(α) = 0. The ﬁrst assertion comes from the above identities and dim X = n. QED k We set Θr to be the sheaf generated (in the sense of [29]) by the presheaf ker h + d ker h. Remark 2.1. We stress that, in general, the sheaf axioms fail to be true for d ker h. Anyway, if dim X = 1 and k > 1, the sum ker h + d ker h turns out to be a direct sum [26], and d ker h turns out to be a sheaf. In the rest of this section, we also denote by d ker h the sheaf generated by the presheaf d ker h, by an abuse of notation.

k k Lemma 2.2. If 0 ≤ k ≤ n, then d ker h ⊂ ker h, so that Θr = CΛr. Proof. By the above Lemma, if α ∈ ker h, then for any section s : X → Y we ∗ ∗ have (jrs) α = 0, hence (jrs) dα = 0. So, dα ∈ ker h. QED k k It is clear that Θr is a subsheaf of Λr. Thus, we say the following natural subsequence

1 2 I - d - d - d - d - 0 Θr Θr . . . Θr 0 to be the contact subsequence of the de Rham sequence. We note that, in general, the k ∗ sheaves Θr are not the sheaves of sections of a vector subbundle of T JrY .

Remark 2.2. In general, I depends on the dimension of the ﬁbers of JrY → X; its value is given in [10]. R. Vitolo 15

The following theorem is proved in [10].

Theorem 2.1. The contact subsequence is exact and soft.

Now, we introduce a bicomplex by quotienting the de Rham sequence on JrY by the contact subsequence. We obtain a new sequence, the variational sequence, which turns out to be exact. In the last part of the section, we describe the relationships between bicomplexes on jet spaces of diﬀerent orders.

Proposition 2.1. The following diagram 0000 00

? ? ? ? ? 1 2 I ? - - - d - d - d - d - - - 0 0 0 Θr Θr . . . Θr 0 . . . 0

? ? ? ? ? ? 0 1 2 I I+1 - - d- d - d - d - d- d- -d 0 IR Λr Λr Λr . . . Λr Λ r . . . 0

? ? ? ? ? ? 0 1 1 2 2 I I I+1 - - E-0 E-1 E-2 EI−-1 E-I d- -d 0 IR Λr Λr/Θr Λr/Θr . . . Λr/Θr Λ r . . . 0

? ? ? ? ? ? 0 0 0 0 0 0 is a commutative diagram whose rows and columns are exact. Proof. We have to prove only the exactness of the bottom row of the diagram. But this follows from the exactness of the other rows and of the columns.

Deﬁnition 2.1. We say the bottom row of the above diagram to be the r–th order variational sequence associated with the ﬁbred manifold Y → X (see [10]).

We stress that this sequence is obtained in an intrinsic way, but it is not the unique intrinsic one. It is obtained in order to match precise criteria, i.e. to obtain an exact sheaf sequence that carries the appropriate information for the calculus of variations.

k k Proposition 2.2. The sheaves Λr/Θr are soft sheaves [10].

k k Proof. In fact, each column is a short exact sheaf sequence in which Θr and Λr are soft sheaves (see [29]). QED

Corollary 2.1. The variational sequence is a soft resolution of the constant sheaf IR over Y [10]. 16 On formulations of Lagrangian formalism

Proof. In fact, except IR, each one of the sheaves in the sequence is soft [29]. The most interesting consequence of the above corollary is the following one (for a proof, see [29]). Let us consider the cochain complex

0 1 1 J - - d- E1 - d - d - 0 IRY Λr Λr/Θr . . . Λr 0 Y Y Y

k and denote by HVS the k–th cohomology group of the above cochain complex.

Corollary 2.2. For all k ≥ 0 there is a natural isomorphism

k k HVS ≃ Hde RhamY

(see [10]). Proof. In fact, the Lagrangian sequence is a soft resolution of IR, hence the cohomology of the sheaf IR is naturally isomorphic to the cohomology of the above cochain complex. Also, the de Rham sequence gives rise to a cochain complex of global sections, whose cohomology is naturally isomorphic to the cohomology of the sheaf IR on Y . Hence, we have the result by a composition of isomorphisms. (See [29] for more details on the above natural isomorphisms.) QED

3 Finite order C–spectral sequence

The C–spectral sequence has been introduced by Vinogradov [23, 24, 25]. It is a very powerful tool in the study of differential equations. Here, we present a new finite order approach to variational sequences by means of ∗ the C–spectral sequence induced by the de Rham exact sequence (Λr,d) (see Lemma 8.1) on the jet space of order r of a fibred manifold. It shall be remarked that such an approach has already been attempted in a very particular case [5]. Indeed, our finite order formulation presents some technical difficulties: our main tool is the splitting of Theorem 1.2, where the direct summands have a rather complicated structure and, k above all, are not subsheaves of Λr. Then, we show the correspondence between the simplified finite order variational sequence and the variational sequence obtained via the finite order C–spectral sequence.

Remark 3.1. The ﬁnite order C–spectral sequence is formulated here in the category of presheaves of IR–vector spaces. This means that the constructions of Appendix B will be done on any open set. We stress that the reason for doing this a lie in the fact that, in our case, the function mapping open sets into homology groups is not a sheaf, but just a presheaf. R. Vitolo 17

∗ We consider the sheaf of diﬀerential groups (Λr,d) and the graded sheaf ﬁltration ∗ p (C Λr,d)p∈N, where

∗ ∗ ∗ 1 ∗ C Λr := CΛr ≡ {α ∈ Λr | ∀ s section of Y → X (jrs) α = 0}

∗ ∗ ∗ ∗ ∗ k p 1 0 p and C Λr is the p–th power of the ideal C Λr in Λr. We set C Λr = Λr, and C Λr = {0} if p > k. We recall that

p+q p+q p,q p p+1 E0 ≡ C Λ r C Λ r . Moreover, we recall the exact sequence of Lemma 8.3. p+q p As a preliminar step, we look for a description of the sheaves C Λ r. To this aim, we introduce new projections associated to the splitting of proposition 1.1 Let 0 ≤ q ≤ n; we denote by Hp the projection

p p+q p−l q+l H (r+1,r) → C (r+1,r) ∧ H r+1 ; Ml=1 we denote by V p the complementary projection, i.e. V p = id − Hp. Of course, Hp = 0 if q = n. Also, we denote by hp and vp the corresponding restrictions to the subsheaf k Λr. Lemma 3.1. We have

H1 = H, Hp = H if q = n − 1 .

Remark 3.2. By the above lemma, if p> 1 and q < n − 1 then hp is not surjective on p−l q+l p h ⊕l=1 C r ∧ H r+1, in general. But the most interesting cases are p = 1 and q = n − 1, where hp = h is surjective. Lemma 3.2. Let p ≥ 1. Then, we have

p+q p p C Λ r ≃ ker h if q < n ; p+q p+q p C Λ r = Λ r if q ≥ n . Proof. We recall that (lemma 2.1 and lemma 3.1) the theorem holds for p = 1. Then, we have the identities ker Hp = im V p and im V p = h(im V )pi = h(ker H)pi, where h(im V )pi denotes the ideal generated by pth exterior powers of elements of im V 1+q k p p in Λ r. So, by restriction to Λr, we have ker h = h(ker h) i. But, by deﬁnition and p+q p p lemma 2.1, we have C Λ r = h(ker h) i, hence the result. QED

Now, we compute (E0,e0). 18 On formulations of Lagrangian formalism

Lemma 3.3. We have

p,0 p E0 = ker h ; p q p,q h E0 ⊂ Cr ∧ Hr+1 if 1 ≤ q < n ; p n p,n h E0 ≃ Cr ∧ Hr+1 p,q E0 = {0} otherwise; ¯ p,q p,q p,q+1 p+1 p+2 d ≡ e0 : E0 → E0 : h (α) 7→ h (dα) . Proof. The ﬁrst and fourth assertions are trivial. As for the second one, the inclusion is realised via the injective morphism

p q p,q p p+1 h p+1 E0 ≡ ker h ker h → Cr ∧ Hr+1 : [α] 7→ h (α) . p+n The third statement comes from the identity hp = 0 if q = n, which imply ker hp = Λ , and lemma 3.1, which imply that hp+1 is surjective. The sheaf morphism d¯ can be read through the above morphism; we obtain the last assertion. QED

Proposition 3.1. The bigraded complex (E0,e0) is isomorphic to the sequence of cochain complexes 0 0 0 . . . 0

? ? ? ? 0 1 2 I Λr ker h ker h . . . ker h

d¯ ? −d¯ d¯ (−1)I d¯ ? ? ? 1 h 1,1 2,1 I,1 Hr E0 E0 ... E0

d¯ −d¯ d¯ (−1)I d¯ ? ? ? ? ......

¯ ¯ ¯ I ¯ d ? −d ? d ? (−1) d ? n 1 n 2 n I n h h h h Hr Cr ∧ Hr+1 Cr ∧ Hr+1 . . . Cr ∧ Hr+1

d¯ −d¯ d¯ (−1)I d¯ ? ? ? ? 0 0 0 0 The sequence becomes trivial after the I–th column. The minus signs are put in order to agree with an analogous convention on inﬁnite order variational bicomplexes. R. Vitolo 19

Remark 3.3. As we will see, in the infinite order case the sheaf morphism dv yields horizontal arrows in the sequence analogous to the above one. So, we obtain a commutative bicomplex. Here, we have no horizontal arrows, due to the fact that the maps dv raise the order of the jet by one. Another difference with the infinite order sequence is that here the sequence becomes trivial after a certain value of the degree p.

We note that the bottom row of the above sequence projects to 0. Also, we recall that E1 = H(E0), where the homology is taken with respect to the sheaf morphism dh. These two facts yield the following Theorem.

Theorem 3.1. We have the bicomplex

0 0

? 0 ? p Λr . . . ker h . . .

d¯ ? (−1)I d¯ ? 1 h p,1 Hr ... E0 . . .

d¯ (−1)I d¯ ? ? ......

¯ I ¯ d ? (−1) d ? n p n h h Hr . . . Cr ∧ Hr+1 . . .

I d¯ ? (−1) d¯ ? n n−1 ′ E′ p n E′ - h ¯ h En - n+p−1- h ¯ p,n−1 n+p- 0 Hr d( H r ) . . . (Cr ∧ Hr+1) d(E0 ) . . . ? ? 0 0 where the bottom row is a presheaf of cochain complexes. The bicomplex is trivial if p>I and vertical arrows with values into the quotients are trivial projections. We 20 On formulations of Lagrangian formalism have the identiﬁcations

n n−1 0,n h ¯ h E1 = Hr+1 d( H r+1) , p n p n−1 p,n h ¯ h E1 =(Cr ∧ Hr+1) d(Cr ∧ H r+1) , n n−1 1 n−1 0,n ′ h ¯ h 1,n−1 ¯ h e1 = En : Hr+1 d( H r+1) → (E0 ) d(Cr ∧ H r+1) : [h1(α)]7→ [h2(dα)] , p n p+1 n p,n ′ h ¯ p,n−1 h ¯ p+1,n−1 e1 = Ep+n :(Cr ∧ Hr+1) d(E0 ) → ( C r ∧ Hr+1) d(E0 ) : [hp+1(α)] 7→ [hp+2(dα)] .

Proof. The above identifications come directly from the definition of E1. As for the last statement, by recalling the exact sequence of Lemma 8.3, we have by definition

p,1 e1 = π◦δ , where δ is the Bockstein operator induced by the exact sequence and π is the cohomology map induced by the corresponding map π of the exact sequence. So, suppose that

p n p+1 p,n h h (α) ∈ E0 = Cr ∧ Hr+1 ;

p+n we have α ∈ Λ r. Then,

π(dα)= d¯(π(α))=0 ,

p+1+n p+1 because d¯ raises the degree by 1 on the horizontal factor, so, dα ∈ C Λ r. Being p+1+n p+1 d(dα) = 0, dα is closed in C Λ r under the diﬀerential d, but dα is not exact in p+n p+n p+1 p+1 p+1 C Λ r, i.e. there does not exist a form β ∈ C Λ r = ker h such that dβ = α. p+1+n p+1 Hence, dα determines a cohomology class [dα] in C Λ r which is, by deﬁnition, the value of δ([hp+1(α)]). The map π maps dα into hp+2(dα), hence the cohomology class p [dα] is mapped into [h +2(dα)] by π. QED R. Vitolo 21

Theorem 3.2. We have the commutative diagram 0

? 0 Λr

d¯ ? 1 h Hr

d¯ ? . . .

¯ d ? n Hh r ˜ En - ? n n−1 ′ 1 n E′ - h ¯ h En- h ¯ 1,n−1 n+1 - 0 Hr d( H r ) (Cr ∧ Hr+1) d(E0 ) . . . ? 0 ˜ ′ ¯ where En := En ◦ d, and the sequence

˜ n ˜ 1 n E′ En−1 - h En - h ¯ 1,n−1 n+1- . . . Hr (Cr ∧ Hr+1) d(E0 ) . . . is a complex of presheaves. Definition 3.1. We say the bottom row of the above bicomplex to be the finite order variational sequence associated with the finite order C–spectral sequence on Y → X.

The cohomology of the above sequence will be clear in next section after proving that it is isomorphic with the ﬁnite order variational sequence of deﬁnition 2.1.

4 Comparison between ﬁnite order approaches

In this section, we show the connection between Krupka’s variational sequence and the variational sequence associated with the ﬁnite order C–spectral sequence. First of all, we provide a simpliﬁed version of Krupka’s variational sequence , i.e. , a sequence which is isomorphic to Krupka’s variational sequence but is made by sheaves of forms or by quotient sheaves which are quotients between ‘smaller’ sheaves. In the case 0 ≤ k ≤ n, lemma 2.2 yields immediately the following result. 22 On formulations of Lagrangian formalism

Theorem 4.1. Let 0 ≤ k ≤ n. Then, the sheaf morphism h yields the isomorphism

k k k h Ik : Λr/Θr → Hr+1 : [α] 7→ h(α) .

In the case k > n, we are able to provide isomorphisms of the quotient sheaves with other quotient sheaves made with proper subsheaves.

Proposition 4.1. Let k > n. Then, the projection h induces the natural sheaf isomorphism

k k k−n n Λ /Θ → C ∧ Hh h(d ker h) : [α] 7→ [h(α)] , r r r r+1 where d ker h stands for the sheaf generated by the presheaf d ker h, by an abuse of notation. Proof. The map is clearly well deﬁned. k ′ Also, the map is injective, for if α, α ∈ Λr, then

[h(α)]=[h(α′)] ⇒ h(α − α′)= hdp , with p ∈ ker h. Hence

α − α′ = v(α − α′ − dp)+ dp ,

k k k ′ ′ where, being dp ∈ Λr and α − α ∈ Λr, we have v(α − α − dp) ∈ Λr. Due to h ◦ v = 0, we have [α − α′] = 0. Finally, the map is surjective, due to the surjectivity of h. QED

r Remark 4.1. Let 0 ≤ s ≤ r. Then, the sheaf injection χs induces the sheaf injection

k−n n k−n n C ∧ Hh h(d ker h) → C ∧ Hh h(d ker h) . s s+1 r r+1 Theorem 4.2. Krupka’s r–th order variational sequence is isomorphic to the sequence

0 1 n - E0 - h E1 - En−1- h En - 0 Λr Hr . . . Hr

1 n E E − i n E C ∧ Hh h(d ker h) n+1- . . . n+i -1 Ch ∧ Hh h(d ker h) n+-i . . . r r+1 r r+1 where E0 coincides with dh, and Ek([h(α)]) = [h(dα)]. Hence Krupka’s variational sequence is isomorphic to the variational sequence associated with the ﬁnite order C- spectral sequence, which turns out to be exact. R. Vitolo 23

Theorem 4.3. (First comparison theorem). We have the identiﬁcations

Ek = d¯ , 0 ≤ k

Proof. It comes from the above theorem and the deﬁnition of d¯. QED

Theorem 4.4. (Second comparison theorem). We have the identiﬁcations ¯ p,n−1 d(E0 )= h(d ker h) ,

E˜n(h(α))=[h(dα)] , ′ Ek(h(α))=[h(dα)] , n

¯ p,n−1 p+1 p d(E0 )= h (d ker h )

p+1+n−1 p p+1 ¯ p,n−1 but h = h being q = n − 1, and h = h on Λ r, hence d(E0 )= h(d ker h). For the two others, we use lemma 3.1, and observe that En ◦ d¯ = En ◦ En−1 = 0, so n−1 ¯ h QED d( H r ) ⊂ ker En, hence the result follows. The above results prove that the two formulations yield the same variational sequence up to the degree n. Indeed, we can improve this result and state the equivalence up to the order n + 1. Proposition 4.2. We have the sequence of presheaf isomorphisms

1 n 1 n 1 n C ∧ Hh h(d ker h) ≃ C ∧ Hh + P ∩ C ∧ H ≃ r r+1 r r+1 (2r+1,0) 2r+1 1 n 1 n−1 1 n 1 n h h p,n−1 C ∧ H + d (C − ∧ H ) ∩ C ∧ H ≃ C ∧ H d¯(E ) , r r+1 h (2r,r 1) 2r (2r+1,0) 2r+1 r r+1 0 where d ker h stands for the sheaf generated by d ker h and P stands for the sheaf gen- 1 n−1 erated by dh(C(2r,r−1) ∧ H 2r). Proof. The first isomorphism is proved in [28], and it is built essentially by means of the first variation formula, as given in [9]. The first variation formula yields a section 1 n h of P for any section of Cr ∧ Hr+1, but, as it is shown in [9], such a section is indeed 1 n−1 a section of the presheaf dh(C(2r,r−1) ∧ H 2r) generating P, i.e. , it is of globally of the 1 n−1 form dhp, with p ∈ C(2r,r−1) ∧ H 2r. Hence, the second isomorphism holds. The last isomorphism is obtained in the same way of the first one. QED 24 On formulations of Lagrangian formalism

Corollary 4.1. We have the identification Eñ = En. By recalling the intrinsic interpretation in terms of the calculus of variations of the variational sequence given in [28], we give the following definition.

Deﬁnition 4.1. We say each one of the sheaves of proposition 4.2 to be the sheaf of Euler–Lagrange morphisms.

Remark 4.2. It is very important to note that Krupka’s formulation could be modified by using as the contact subsequence the presheaf ker h + d ker h. This would yield an exact finite order variational sequence (exactness is a local matter), with the unique drawback of the impossibility of computing its cohomology with the de Rham theorem from sheaf theory. But the sequence obtained in this way would be exactly equal to the variational sequence obtained with the finite order C–spectral sequence. And the cohomology of the last sequence has been compute above!

5 Inﬁnite order variational sequence

In this section, we analyse the relationships between Krupka’s finite order variational bicomplexes of different orders. In particular, we provide a natural inclusion of the variational bicomplex of order s > 0 into each variational bicomplex of order r>s. Then, we evaluate the direct limit of the system of bicomplexes, obtaining an infinite order variational sequence as the direct limit of the injective system of the finite order variational sequences. As far as we know, this approach is original. We have the injective system of sheaves

k r∗ {Θs,πs } .

r∗ Lemma 5.1. [10]. Let s ≤ r. Then, the injective sheaf morphism πs induce the injective sheaf morphism

k k k k χr : Λ /Θ → Λ /Θ : [α] 7→ [πr∗α] . s s s r r s Proof. r The above morphism χs is well deﬁned, because

r∗ r∗ [α] = [β] ⇒ [πs α] = [πs β] .

k k The above morphism is also injective, for if α ∈ Λs and β ∈ Λs are such that

r∗ r∗ [πs α] = [πs β] ,

k k k r∗ r∗ r∗ r∗ r∗ then, being πs (α − β) ∈ πs Λs, and πs (α − β) ∈ Θr, we have πs (α − β) ∈ πs Θs, hence [α] = [β]. QED R. Vitolo 25

Proposition 5.1. We have the injective system of sheaves

k k r {(Λr/Θr) ,χs} .

Remark 5.1. We have the commutative diagrams

k k+1 k k+1 d - d - Λr Λ r Θr Θ r 6 6 6 6 r∗ r∗ r∗ r∗ πs πs πs πs

k k+1 k k+1 d - d - Λs Λ s Θs Θ s hence we have the commutative diagram

k k k+1 k+1 E-k Λr/Θr Λ r/ Θ r 6 6 r∗ r∗ χs χs

k k k+1 k+1 E-k Λs/Θs Λ s/ Θ s

We can summarise the above result by stating the existence of a (non exact) three– dimensional commutative diagram, whose bidimensional slices are the variational bicomplexes.

We deﬁne the presheaves on Y

k k Θ := lim Θr . →

Lemma 5.2. We have

k k k k Λ Θ = lim Λr/Θr . →

Lemma 5.3. The sheaf morphisms Ek induce the presheaf morphisms

k k k+1 k+1 E : Λ/Θ → Λ / Θ : [α] 7→ [dα] , k

k for each k ≥ 0, where, being α ∈ Λr for some r, dα coincides with the diﬀerential of α k on Λr. 26 On formulations of Lagrangian formalism

Theorem 5.1. The following diagram

0000 0

? ? ? ? ? 1 d 2 d d k d 0 - 0 - 0 - Θ - Θ - . . . - Θ - . . .

? ? ? ? ? 0 d 1 d 2 d d k d 0 - IR - Λ - Λ - Λ - . . . - Λ - . . .

? ? ? ? ? 0 1 1 2 2 k k - - E0 - E1 - E2- Ek−1 - Ek- 0 IR Λr Λ/Θ Λ/Θ . . . Λ/Θ . . .

? ? ? ? ? 0 0 0 0 0 is commutative, and rows and columns are exact presheaf sequences.

Proof. By the analogous result for ﬁnite order variational bicomplexes. QED

Note that E0 coincides with dh. Moreover, the diagram does not become trivial after a certain value of k, as in the ﬁnite order case.

Deﬁnition 5.1. The bottom row of the above diagram is said to be the inﬁnite order variational sequence.

6 Inﬁnite order C–spectral sequence

In this section we show that the above infinite order variational sequence can be recov- ered by means of the C–spectral sequence arising naturally from a fibred manifold (see the Appendix B) [23, 24, 25]. Indeed, we show that the C–spectral sequence induced ∗ by the de Rham exact sequence (Λ,d) (see Lemma 8.1) allows us to recover the infinite order variational sequence. We recall that the C–spectral sequence is the spectral sequence associated with the ∗ ∗ p cochain complex (Λ,d) and the graded filtration (C Λ,d)p∈N, where

∗ ∗ C1Λ := {ϑ ∈ Λ | ∀ s section of Y → X (js)∗ϑ = 0}

∗ ∗ ∗ ∗ k k and CpΛ is the p–th power of the ideal C1Λ in Λ. We set C0Λ = {0}, and CpΛ = Λ if p > k. R. Vitolo 27

k p s Remark 6.1. We have the injective systems {C Λs,πr}, and

k k p p C Λ = lim C Λs . →

Hence, the computations of the inﬁnite order C–spectral sequence can be performed both by direct evaluation and by direct limit. We will devote little space to proofs in the inﬁnite order case; the interested reader can consult [23, 24, 25].

A version of Lemma 3.2 can be given in the inﬁnite order case. Hence, we can ∗ describe the presheaves CpΛ. The splitting of Theorem 1.3 yields the result in a much simpler way, with respect to the ﬁnite order case.

Lemma 6.1. Let p ≥ 1. Then, we have

p q p+q p C∧ H if 0 ≤ q ≤ n − 1 ; C Λ = p+q  Λ if q ≥ n .  Lemma 6.2. [25, p.72] We have

p q p,q p,q E0 = C∧ H , if 0 ≤ q ≤ n ; E0 = {0} otherwise; p q p q+1 p,q e0 = dh : C∧ H → C∧ H .

In other words, the bigraded complex (E0,e0) coincides with the bigraded complex ∗ ∗ (C∧ H,dh), where the bigrading is given by the splitting of Theorem 1.3. Anyway, this splitting yields the presheaf morphism dv too.

∗ ∗ Proposition 6.1. The bigraded complex (C∧H,dv) yields a bigraded complex structure ∗ on C∗Λ via the equalities of Corollary 6.1. More precisely,

p q p+1 q dv : C∧ H → C ∧ H .

We recall that E1 = H(E0), where the homology is taken with respect to the presheaf morphism dh. These two facts yield the following Theorem.

∗ ∗ ∗ ∗ Theorem 6.1. The above bigraded complexes (C ∧ H,dh) and (C ∧ H,dv) yield the 28 On formulations of Lagrangian formalism bicomplex 0 0 0

? ? ? 0 d 1 d 2 d 0 - Λ v - C v - C v - . . .

dh ? −dh ? dh ? 1 d 1 1 d 2 1 d 0 - H v - C∧ H v - C∧ H v - . . .

dh −dh dh ? ? ? ......

? ? ? n 1 n 2 n H e0,n C∧ H e1,n C∧ H e2,n 0 - 1 - 1- 1 - . . . n−1 2 n−1 2 n−1 dh H dh(C∧ H ) dh(C∧ H ) ? ? ? 0 0 0 which contains the direct limit of the finite order bicomplex arising from the C–spectral sequence on finite order jets. We have the identifications

p n p+1 n−1 p,n E1 =(C∧ H) dh( C ∧ H ) , p,n ′ e1 = Ep+1 Proof. It is easy to see that the direct limit of the ﬁnite order bicomplex induced by the C–spectral sequence is constituted by the columns of the above bicomplex together with the bottom row. In particular, one can see that the above presheaf morphisms p,n QED e1 are the direct limit of the corresponding ones of the ﬁnite order case.

7 Comparison between inﬁnite order approaches

We evaluate the direct limit of the simpliﬁed version of the variational sequences of order r, given in theorem 4.2. Clearly, this limit turns out to be isomorphic to the direct limit of ﬁnite order variational sequences. Remark 7.1. Let 0 ≤ s ≤ r. Then, by recalling the injective morphism of remark 4.1, we have the injective system of sheaves

k k−n n {Hh,πr} if 0 ≤ k ≤ n , { C h ∧ Hh h(d ker h) ,χr} if n

Lemma 7.1. The following inclusions hold k−n n−1 k h h(d ker h) ⊂ dh( C r ∧ H r+1) ⊂ h(Θr+1)= h(d ker h) . Proof. By using the decomposition d = dh + dv. QED Proposition 7.1. Let k > 1. Then, we have the natural isomorphisms k k k (6) Λ/Θ ≃ H , if 0 ≤ k ≤ n k k k−n n k−n+1 n−1 (7) Λ/Θ ≃ C ∧ H d ( C ∧ H ) if n

0 E 1 E E − n E 0 - Λ 0 - H 1 - . . . n 1 - H n - 1 n 2 n−1 i n i+1 n−1 En+1- En+i−1- En+i- (C∧ H) dh(C∧ H ) . . . (C∧ H) dh( C ∧ H ) . . . where Ek coincides with dh if 0 ≤ k ≤ n, and Ek([α]) = [dv(α)] if k > n. Proof. In fact, the isomorphisms (7) come from the above lemma. We have to prove that Ek([α]) = [dv(α)]. But we have α = h(β), and

Ek([α]) = Ek([h(β)]) = [h(dβ)] , with

h(dβ)= h((dh + dv)(h(β)+ v(β)) = dv(h(β)) + dh(v(β)) , hence the result. QED Thus, we have provided the infinite order analogue (indeed, the direct limit) of the sequence of Theorem 4.2. As for the comparison between the above sequence and the infinite order variational sequences associated with the C–spectral sequence, we note that the results of theorems4.3, 4.4 and proposition 4.2, and even remark 4.2 hold in the direct limit. Theorem 7.1. The infinite order variational sequence provided by the direct limit of Krupka’s variational sequence and the infinite order variational sequences associated with the C–spectral sequence are isomorphic up to the degree n + 1. In particular, the space of infinite order Euler–Lagrange morphism turn out to be 1 n 1 1 C(∗,0) ∧ H, where C(∗,0) := lim C(r,0). → Proof. The first part comes from the above quoted results, and the last assertion comes from the following inclusions 1 n 1 n C ∧ Hh + P ∩ C ∧ H ⊂ r r+1 (2r+1,0) 2r+1 1 n ⊂ C(2r+1,0) ∧ H2r+1 ⊂ 1 n 1 n h ⊂ C ∧ H + P ∩ C ∧ H . QED 2r+1 2r+2 (4r+3,0) 4r+3 30 On formulations of Lagrangian formalism

8 Conclusions

We have shown what are the relationships between two of the most important geometric formulations of Lagrangian formalism. Moreover, we provided two new formulations, each of which is inspired by one of the above two. We stress that each formulation can be carried on independently, giving rise to two exact sequences with the same cohomologies, and any of the two yields the same information for the Lagrangian formalism up to the degree n + 1. As for the degree n + 2, we recall [10] that this yields information on the local variationality of Euler–Lagrange operators, and there exists an intrinsic formuations of the conditions of local variationality (Helmholz morphism, see [28]). At the present moment we are studying the equivalence of the sequence at the degree n + 2, and we found an equivalence up to the order r = 2. We stress that there is no interpretation in terms of the geometric objects of the calculus of variations for sections having degree k > n + 2.

Appendix A: direct sums and exterior products

Let V be a vector space such that dim V = n. We recall that the box product (see, for example, [7]) of r linear morphisms a1,...ar : V → V is deﬁned to be the linear map

r r a1 . . . ar : ∧ V → ∧V :

v1 ∧ . . . ∧ vr 7→ |σ|a1(vσ(1)) ∧ . . . ∧ ar(vσ(r)) . σX∈Sr where Sr is the set of all permutation of order r. The box product fulﬁlls

a1 . . . ar = aσ(1) . . . aσ(r) ∀ σ ∈ Sr , r a . . . a = r! ∧ a ;

k k so, yields a map ⊙(End(V )) → End(∧V ). We have a remarkable feature of the box product. Suppose that V = W1 ⊕W2, with p1 : V → W1 and p2 : V → W2 the related projections. Then, we have the splitting

m k h (8) ∧ V = ∧W1 ∧ ∧W2 , k+Mh=m

k h m where ∧W1 ∧ ∧W2 is the subspace of ∧V generated by the wedge products of elements k h of ∧W1 and ∧W2. The projections pk,h related to the above splitting turn out to be the maps

1 m k h p = kp hp : ∧V → ∧W ∧ ∧W . k,h k! h! 1 2 1 2 R. Vitolo 31

′ ′ ′ ′ ′ Remark 8.1. Let V ⊂ V be a vector subspace, and set W1 := p1(V ), W2 := p2(V ). Then we have

′ ′ ′ V ⊂ W1 ⊕ W2 , but the inclusion, in general, is not an equality.

Appendix B: spectral sequences

In this section, we give the basic material on spectral sequences. In the first subsection we recall the definition of spectral sequence, togheter with some preliminar concepts. In the second subsection we give the notions of exact couple and derived couple. The third subsection is devoted to the definition of spectral sequence associated with a filtration of a given complex. The interest reader can consult [3, 12, 14, 18] for more details and applications.

Spectral sequences In this subsection we give some preliminar deﬁnitions. Note that we will introduce graded groups and maps with degrees in N rather than Z. This is due to the fact that in our applications we will not need a grading in Z.

Definition 8.1. A differential group is defined to be a pair (Λ,d), where Λ is an Abelian group and d : Λ → Λ is a group morphism such that d2 = 0, or, equivalently, im d ⊂ ker d. The morphism d is said to be the differential of Λ. The homology of the differential group is defined to be the abelian group

H(Λ) := ker d im d . ∗ Definition 8.2. A graded differential group (of degree g) is defined to be a pair (Λ,d), where

∗ k Λ := ⊕k∈N Λ

∗ ∗ is a graded Abelian group and d : Λ → Λ is a graded morphism of degree g, i.e.

k k+g d(Λ) ⊂ Λ , such that d2 = 0, or, equivalently, im d ⊂ ker d.

We recall that a cochain complex is a sequence of morphisms of abelian groups of the form 0 d 1 d 2 d 0 - Λ 0 - Λ 1 - Λ 2 - . . . 32 On formulations of Lagrangian formalism

such that dk+1◦dk = 0. This last condition is equivalent to im dk ⊂ ker dk+1. A cochain complex is said to be an exact sequence if im dk = ker dk+1. To each cochain complex we can deﬁne the cohomology group

∗ ∗ ∗ k H (Λ) = ⊕k∈NH (Λ) , where ∗ k H (Λ) := (ker dk) (im dk−1) . The cohomology groups vanish if and only if the cochain complex is an exact sequence.

∗ Lemma 8.1. There is a bijection between graded diﬀerential groups (Λ,d) of degree +1 and cochain complexes

0 d 1 d 2 d 0 - Λ - Λ - Λ - . . . ∗ Moreover, the homology of (Λ,d) coincides with the cohomology of the corresponding cochain complex.

∗ So, we identify any graded differential group (Λ,d) of degree +1 with the cochain ∗ complex associated with (Λ,d) via the above Lemma. Definition 8.3. We define a spectral sequence to be a sequence of differential groups

(En,en)n∈N such that

En+1 = H(En) .

We say that the spectral sequence converges to Er if Er = Ek for any k > r.

Exact couples Deﬁnition 8.4. An exact couple is deﬁned to be a pair (Q, S) of abelian groups togheter with an exact sequence of morphisms

i - S S

δ π

Q R. Vitolo 33

Remark 8.2. If (Q, S) is an exact couple as above, then the pair (Q, e), where e := π◦δ, is a diﬀerential group. In fact, (π ◦ δ)2 = 0 due to the exactness of the above diagram.

Proposition 8.1. Let (Q, S) be an exact couple, as in the above deﬁnition. Then, the pair (E1,S1), where

E1 := H(Q) , S1 := i(S) , together with the diagram

i1 - S1 S1

δ 1 π 1

Q1 where

i1 : i(S) → i(S) : i(s) 7→ i(i(s)) ,

π1 : i(S) → H(Q) : i(s) 7→ [π(s)] ,

δ1 : H(Q) → i(S) : [q] 7→ δ(q) , is an exact couple.

Proof. One has to check that the above maps are well deﬁned, and that the above diagram is commutative and exact. This is straightforward. QED

The above exact couple is said to be the derived couple. The pair (E1,e1), where e1 := π1 ◦ δ1, turn out to be a diﬀerential group. We can consider iterated derived couples; namely, we set by induction

(E0,S0) := (E,S) ;

(En+1,Sn+1) := ((En)1, (Sn)1) ∀ n> 0 ; analogously, we deﬁne in,πn, δn,en. So, we have the sequence of diﬀerential groups (En,en)n∈N, and the following obvious result.

Proposition 8.2. Any exact couple (E, Q) yields a spectral sequence (En,en)n∈N.

Remark 8.3. We remark that, if Q and S are graded abelian groups, i, π are graded morphisms of degree 0 and δ is a graded morphism of degree +1, then (En,en)n∈N is a spectral sequence which is made by graded diﬀerential groups. 34 On formulations of Lagrangian formalism

Filtered differential groups Let (Λ,d) be a differential group. A differential subgroup is defined to be a differential ′ ′ ′ group (C,d ) such that C ⊂ Λ is an abelian subgroup and d = d|C. We will denote d by d, by an abuse of notation. Definition 8.5. We define a filtration of a differential group (Λ,d) to be a sequence p 0 of differential subgroups (C ,d)p∈N of (Λ,d), where C := Λ, which is decreasing with respect to the inclusion, namely Λ ≡ C0 ⊃ C1 ⊃ C2 ⊃ . . . If there exists l ∈ N such that Cl =6 {0} but Ck = {0} for k > l, then we say that the filtration has finite length l. p If (C ,d)p∈N is a filtration of (Λ,d), then we say (Λ,d) to be a filtered differential group. p Let (C ,d)p∈N be a filtration of (Λ,d). We define the abelian groups p p p+1 p Q := C C , Q := ⊕p∈N Q . Lemma 8.2. For each p ≥ 0, the morphism d passes to the quotient Cp/Cp+1. The induced graded morphism of degree 0 d¯ : Q → Q fulfills d¯2 = 0. Hence, we have the graded differential group (of degree 0) (Q, d¯). The pair (Q, d¯) is said to be the graded differential group associated with the filtration. Moreover, we define the graded differential group (of degree 0) p S := ⊕p∈N C . Lemma 8.3. We have the graded exact sequence of graded differential groups i π 0 - S - S - Q - 0

p+1 p where i|Cp+1 : C → C is the inclusion map, of degree −1, and p p p+1 π|Cp : C → C C is the natural projection, of degree 0. The maps i, π commute with the diﬀerentials in the domains and codomains. Passing to cohomologies, we obtain the exact sequence i π δ . . . - Hk(S) - Hk(S) - Hk(Q) - Hk+1(S) - . . . which yields the exact couple ∗ i - ∗ H (S) H (S)

δ π

H∗(Q) R. Vitolo 35 where δ is the Bockstein operator, of degree +1, and the graded differential group (H∗(Q),e), where e := π ◦ δ has degree +1, and e2 = 0. We stress that i : H∗(S) → H∗(S) is no longer the inclusion map. Remark 8.4. We recall the definition of the Bockstein operator in this context. Let [α] ∈ Hk(Q). Then, being π surjective, we choose β ∈ Sk ≡ Ck such that π(β) = α. We see that π(dβ) = dπ¯ (β) = 0, hence due to the exactness, there exists a unique γ ∈ Sk+1 = Ck+1 such that i(γ) = dβ (actually, γ = dβ, because i is the inclusion map). Finally, dβ is closed in Sk+1, due to d2β = 0, but it is not exact in Sk+1, hence it determines a class [dβ] ∈ Hk+1(S). We can easily prove that δ : Hk(Q) → Hk+1(S) : [α] 7→ [dβ] is well defined.

p Theorem 8.1. Let (Λ,d) be a diﬀerential group. Then, each ﬁltration (C ,d)p∈N of ∗ ∗ (Λ,d) induces a graded spectral sequence (En,en)n∈N (see Remark 8.3) as follows

E0 := Q, e0 := d¯; ∗ ∗ ∗ ∗ (E1 ,S1 ) := (H (Q),H (S)) , e1 := e ≡ π ◦ δ ; ∗ ∗ ∗ n−1 ∗ (En,Sn) := (H (En),i (H (S))) , en := πn ◦ δn

Note that (E0,S) is not an exact couple, but (E0,e0) is a graded differential group (of degree 1). Definition 8.6. Let (Λ,d) be a differential group with a given filtration. We say ∗ ∗ (En,en)n∈N to be the (graded) spectral sequence associated with the filtered differential group (Λ,d). Remark 8.5. We have an important particular case of filtered differential group. ∗ ∗ p Namely, suppose that (Λ,d) is a graded differential group (of degree +1), and (C ,d)p∈N is a graded filtration, i.e. a filtration by graded differential subgroups whose grading is ∗ compatible with the grading of (Λ,d). ∗ The spectral sequence associated with (Λ,d) is a sequence of bigraded complexes ∗,∗ ∗,∗ (En ,en ). More precisely, we have the bigraded differential groups

p+qp q p q−1p+1 ∗,∗ ∗,∗ S := ⊕p,q∈N C , Q := ⊕p,q∈N C C , ∗,∗ p,q En := ⊕ En , p,q∈N

∗ where p is the ﬁltration degree and p + q is the degree induced by Λ; q is said to be the complementary degree. The morphisms i, π, δ turn out to be bigraded morphisms with bidegrees (−1, +1), (0, 0), (+1, 0) respectively. Moreover, it can be proved that the maps in, πn, δn have bidegrees (1, −1), (n − 1, −n +1), (+1, 0), respectively, hence

p,q p,q p+n,q−n+1 en : En → En . 36 On formulations of Lagrangian formalism

As for the graded case we have a very important result.

∗ Theorem 8.2. Let (Λ,d) be a graded differential group (of degree +1) with a graded ∗ n p p filtration (C ,d)p∈N. Suppose that to any degree n ∈ N the filtration (C ,d)p∈N has finite ∗ length. Then, the spectral sequence induced by the filtration converges to H∗(Λ).

Proof. It can be easily deduced from the deﬁnitions [3, p. 160]. QED

References

[1] I. M. Anderson and T. Duchamp, On the existence of global variational principles, Amer. Math. J., 102 (1980), 781–868.

[2] M. Bauderon, Le probl`eme inverse du calcul des variations, Ann. de l’I.H.P., 36, n. 2 (1982), 159–179.

[3] R. Bott and L. W. Tu, Diﬀerential Forms in Algebraic Topology, GTM 82 (Springer– Verlag, Berlin, 1982).

[4] P. Dedecker and W. M. Tulczyjew, Spectral sequences and the inverse problem of the calculus of variations, in Internat. Coll. on Diﬀ. Geom. Methods in Math. Phys., Aix– en–Provence, 1979; Lecture Notes in Mathematics 836 (Springer–Verlag, Berlin, 1980), 498–503.

[5] S. V. Duzhin, C–spectral sequence on the manifold J 1M, Uspekhi Math. Nauk 38 (1983), 165–166.

[6] M. Ferraris and M. Francaviglia, Global Formalism in Higher Order Calculus of Varia- tions, Diﬀ. Geom. and its Appl., Part II , Proc. of the Conf. (University J. E. Purkyne, Brno, 1984), 93–117.

[7] W. Greub, Multilinear Algebra (Springer–Verlag, 1978).

[8] I. S. Krasil’shchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Non–linear Partial Diﬀerential Equations (Gordon and Breach, New York, 1986).

[9] I. Kol´aˇr, A geometrical version of the higher order Hamilton formalism in ﬁbred manifolds, Jour. Geom. Phys. 1, n. 2 (1984), 127–137.

[10] D. Krupka, Variational sequences on finite order jet spaces, Diff. Geom. and its Appl., Proc. of the Conf. (World Scientific, New York, 1990) 236–254.

[11] B. A. Kuperschmidt, Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism, Lecture Notes in Math. 775: Geometric Methods in Mathe- matical Physics (Springer, Berlin, (1980)) 162–218.

[12] S. Mac Lane, Homology (Springer–Verlag, New York, 1963). R. Vitolo 37

[13] L. Mangiarotti and M. Modugno, Fibered Spaces, Jet Spaces and Connections for Field Theories, Int. Meet. on Geometry and Physics, Proc. of the Conf. (Pitagora Editrice, Bologna, 1983) 135–165.

[14] W. S. Massey, Algebraic topology: an introduction (Springer–Verlag, New York, 1977).

[15] P. J. Olver, C. Shakiban, A Resolution of the Euler Operator, Proc. Am. Math. Soc., 69 (1978), 223–229.

[16] J. F. Pommaret, Systems of Partial Diﬀerential Equations and Lie Pseudogroups (Gor- don and Breach, New York, 1978).

[17] D. J. Saunders, The Geometry of Jet Bundles (Cambridge Univ. Press, 1989).

[18] E. H. Spanier, Algebraic topology (McGraw–Hill, New York, 1966).

[19] F. Takens, Symmetries, conservation laws and variational principles, Lecture Notes in Math. 597 (Springer, Berlin, 1977).

[20] F. Takens, A global version of the inverse problem of the calculus of variations, J. Diﬀ. Geom. 14 (1979), 543–562.

[21] W. M. Tulczyjew, The Lagrange Complex, Bull. Soc. Math. France 105 (1977), 419– 431.

[22] W. M. Tulczyjew, The Euler–Lagrange Resolution, Internat. Coll. on Diﬀ. Geom. Methods in Math. Phys., Aix–en–Provence, 1979; Lecture Notes in Mathematics 836 (Springer–Verlag, Berlin, 1980), 22–48.

[23] A. M. Vinogradov, On the algebro–geometric foundations of Lagrangian ﬁeld theory, Soviet Math. Dokl. 18 (1977), 1200–1204.

[24] A. M. Vinogradov, A spectral sequence associated with a non–linear diﬀerential equa- tion, and algebro–geometric foundations of Lagrangian ﬁeld theory with constraints, Soviet Math. Dokl. 19 (1978), 144–148.

[25] A. M. Vinogradov, The C–Spectral Sequence, Lagrangian Formalism and Conservation Laws I and II, Journal of Mathematical Analysis and Applications 100, n. 1 (1984).

[26] R. Vitolo, Some aspects of variational sequences in mechanics, Diﬀerential Geometry and its Applications, Proceedings of the V Conference, Brno, 1995.

[27] R. Vitolo, Bicomplessi lagrangiani ed applicazioni alla meccanica relativistica classica e quantistica, PhD Thesis, Dep. of Math., Univ. of Florence, 1996.

[28] R. Vitolo, Finite order Lagrangian bicomplexes, Math. Proc. of the Camb. Phil. Soc., to appear 124 n. 3, 1998.

[29] R. O. Wells, Diﬀerential Analysis on Complex Manifolds, GTM 65 (Springer–Verlag, Berlin, 1980).